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      【IOI2018】狼人
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  <ul class="article-tag-list"><li class="article-tag-list-item"><a class="article-tag-list-link" href="/tags/kruskal重构树/">kruskal重构树</a></li><li class="article-tag-list-item"><a class="article-tag-list-link" href="/tags/扫描线/">扫描线</a></li><li class="article-tag-list-item"><a class="article-tag-list-link" href="/tags/树状数组-BIT/">树状数组(BIT)</a></li></ul>

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        <p><strong>题目链接：<a href="https://loj.ac/problem/2865" target="_blank" rel="noopener">2865. 狼人 - LibreOJ</a></strong></p>
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<p>问题转化一下，其实就是问是否存在一个点，能只经过大于等于L的点到达S，只经过小于等于R的点到达T。于是我们 <s>不知道为什么就忽然</s> 考虑kruskal重构树。</p>
<p>对于树1，我们设边(u,v)的权值为<strong>min(u,v)</strong>，然后求出最<strong>大</strong>生成树的kruskal重构树。设从点u出发，在树1中往上跳到最顶部的权值<strong>大于等于</strong>x的点为v，则v的子树中所有点就是u在原图中只经过<strong>大于等于</strong>x的点能到达的所有点。我们记v=jump1(u,x)</p>
<p>对于树2，我们设边(u,v)的权值为<strong>max(u,v)</strong>，然后求出最<strong>小</strong>生成树的kruskal重构树。设从点u出发，在树2中往上跳到最顶部的权值<strong>小于等于</strong>x的点为v，则v的子树中所有点就是u在原图中只经过<strong>小于等于</strong>x的点能到达的所有点。我们记v=jump2(u,x)</p>
<p>这样一来，一组询问就被我们转化为了：是否存在一个点，同时位于树1的jump1(S,L)子树和树2的jump2(T,R)子树中。如果我们将一个点在树1上的dfs序看作它在二维平面上的横坐标，在树2上的dfs序看作纵坐标，那么原问题就变成了问平面上一个矩形中是否存在点，直接扫描线+树状数组即可</p>
<div class="highlight-box" autocomplete="off" autocorrect="off" autocapitalize="off" spellcheck="false" contenteditable="true" data-rel="CPP"><figure class="iseeu highlight /cpp"><table><tr><td class="gutter"><pre><span class="line">1</span><br><span class="line">2</span><br><span class="line">3</span><br><span class="line">4</span><br><span class="line">5</span><br><span class="line">6</span><br><span class="line">7</span><br><span class="line">8</span><br><span class="line">9</span><br><span class="line">10</span><br><span class="line">11</span><br><span class="line">12</span><br><span class="line">13</span><br><span class="line">14</span><br><span class="line">15</span><br><span class="line">16</span><br><span class="line">17</span><br><span class="line">18</span><br><span class="line">19</span><br><span class="line">20</span><br><span class="line">21</span><br><span class="line">22</span><br><span class="line">23</span><br><span class="line">24</span><br><span 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class="line">87</span><br><span class="line">88</span><br><span class="line">89</span><br><span class="line">90</span><br><span class="line">91</span><br><span class="line">92</span><br><span class="line">93</span><br><span class="line">94</span><br><span class="line">95</span><br><span class="line">96</span><br><span class="line">97</span><br><span class="line">98</span><br></pre></td><td class="code"><pre><span class="line"><span class="meta">#<span class="meta-keyword">include</span><span class="meta-string">&lt;bits/stdc++.h&gt;</span></span></span><br><span class="line"><span class="meta">#<span class="meta-keyword">include</span> <span class="meta-string">"werewolf.h"</span></span></span><br><span class="line"><span class="keyword">using</span> <span class="keyword">namespace</span> <span class="built_in">std</span>;</span><br><span class="line"></span><br><span class="line"><span class="keyword">typedef</span> pair&lt;<span class="keyword">int</span>,<span class="keyword">int</span>&gt; pii;</span><br><span class="line"><span class="keyword">const</span> <span class="keyword">int</span> N=<span class="number">400010</span>;</span><br><span class="line"><span class="class"><span class="keyword">struct</span> <span class="title">E</span>&#123;</span><span class="keyword">int</span> u,v;&#125; e[N];</span><br><span class="line"><span class="class"><span class="keyword">struct</span> <span class="title">Event</span>&#123;</span><span class="keyword">int</span> ty,a,b,c;&#125;;</span><br><span class="line"><span class="keyword">int</span> n,m,q,tot,bit[N];</span><br><span class="line"><span class="built_in">vector</span>&lt;Event&gt; g[N];</span><br><span class="line"><span class="built_in">vector</span>&lt;<span class="keyword">int</span>&gt; ans;</span><br><span class="line"></span><br><span class="line"><span class="function"><span class="keyword">inline</span> <span class="keyword">int</span> <span class="title">min</span><span class="params">(<span class="keyword">int</span> x,<span class="keyword">int</span> y)</span></span>&#123;<span class="keyword">return</span> x&lt;y?x:y;&#125;</span><br><span class="line"><span class="function"><span class="keyword">inline</span> <span class="keyword">int</span> <span class="title">max</span><span class="params">(<span class="keyword">int</span> x,<span class="keyword">int</span> y)</span></span>&#123;<span class="keyword">return</span> x&gt;y?x:y;&#125;</span><br><span class="line"><span class="function"><span class="keyword">inline</span> <span class="keyword">bool</span> <span class="title">leq</span><span class="params">(<span class="keyword">int</span> x,<span class="keyword">int</span> y)</span></span>&#123;<span class="keyword">return</span> x&lt;=y;&#125;</span><br><span class="line"><span class="function"><span class="keyword">inline</span> <span class="keyword">bool</span> <span class="title">geq</span><span class="params">(<span class="keyword">int</span> x,<span class="keyword">int</span> y)</span></span>&#123;<span class="keyword">return</span> x&gt;=y;&#125;</span><br><span class="line"></span><br><span class="line"><span class="class"><span class="keyword">struct</span> <span class="title">Kruskal</span></span></span><br><span class="line"><span class="class">&#123;</span></span><br><span class="line">    <span class="keyword">int</span> fa[N],val[N],ch[N][<span class="number">2</span>],go[<span class="number">20</span>][N];</span><br><span class="line">    <span class="keyword">int</span> in[N],out[N],dfc;</span><br><span class="line">    <span class="function"><span class="keyword">int</span> <span class="title">find</span><span class="params">(<span class="keyword">int</span> x)</span></span>&#123;<span class="keyword">return</span> fa[x]==x?x:fa[x]=find(fa[x]);&#125;</span><br><span class="line">    <span class="function"><span class="keyword">void</span> <span class="title">dfs</span><span class="params">(<span class="keyword">int</span> u)</span></span></span><br><span class="line"><span class="function">    </span>&#123;</span><br><span class="line">        in[u]=++dfc;</span><br><span class="line">        <span class="keyword">if</span>(u&gt;n) dfs(ch[u][<span class="number">0</span>]);</span><br><span class="line">        <span class="keyword">if</span>(u&gt;n) dfs(ch[u][<span class="number">1</span>]);</span><br><span class="line">        out[u]=dfc;</span><br><span class="line">    &#125;</span><br><span class="line">    <span class="function"><span class="keyword">void</span> <span class="title">build</span><span class="params">(<span class="keyword">int</span>(*cmp)(<span class="keyword">int</span>,<span class="keyword">int</span>))</span></span></span><br><span class="line"><span class="function">    </span>&#123;</span><br><span class="line">        dfc=<span class="number">0</span>;tot=n;</span><br><span class="line">        <span class="keyword">for</span>(<span class="keyword">int</span> i=<span class="number">1</span>;i&lt;=n;i++) val[i]=i;</span><br><span class="line">        <span class="keyword">for</span>(<span class="keyword">int</span> i=<span class="number">1</span>;i&lt;=n&lt;&lt;<span class="number">1</span>;i++) fa[i]=i;</span><br><span class="line">        <span class="keyword">for</span>(<span class="keyword">int</span> i=<span class="number">1</span>;i&lt;=m;i++)</span><br><span class="line">        &#123;</span><br><span class="line">            <span class="keyword">int</span> u=e[i].u,v=e[i].v;</span><br><span class="line">            u=find(u);v=find(v);</span><br><span class="line">            <span class="keyword">if</span>(u==v) <span class="keyword">continue</span>;</span><br><span class="line">            val[++tot]=cmp(val[u],val[v]);</span><br><span class="line">            fa[u]=fa[v]=go[<span class="number">0</span>][u]=go[<span class="number">0</span>][v]=tot;</span><br><span class="line">            ch[tot][<span class="number">0</span>]=u;ch[tot][<span class="number">1</span>]=v;</span><br><span class="line">        &#125;</span><br><span class="line">        dfs(tot);</span><br><span class="line">        go[<span class="number">0</span>][tot]=tot;</span><br><span class="line">        <span class="keyword">for</span>(<span class="keyword">int</span> k=<span class="number">1</span>;k&lt;=<span class="number">18</span>;k++)</span><br><span class="line">            <span class="keyword">for</span>(<span class="keyword">int</span> i=<span class="number">1</span>;i&lt;=tot;i++)</span><br><span class="line">                go[k][i]=go[k<span class="number">-1</span>][go[k<span class="number">-1</span>][i]];</span><br><span class="line">    &#125;</span><br><span class="line">    <span class="function">pii <span class="title">jump</span><span class="params">(<span class="keyword">int</span> u,<span class="keyword">int</span> x,<span class="keyword">bool</span>(*cmp)(<span class="keyword">int</span>,<span class="keyword">int</span>))</span></span></span><br><span class="line"><span class="function">    </span>&#123;</span><br><span class="line">        <span class="keyword">for</span>(<span class="keyword">int</span> k=<span class="number">18</span>;k&gt;=<span class="number">0</span>;k--)</span><br><span class="line">            <span class="keyword">if</span>(cmp(val[go[k][u]],x)) u=go[k][u];</span><br><span class="line">        <span class="keyword">return</span> pii(in[u],out[u]);</span><br><span class="line">    &#125;</span><br><span class="line">&#125; t1,t2;</span><br><span class="line"></span><br><span class="line"><span class="function"><span class="keyword">inline</span> <span class="keyword">int</span> <span class="title">lowbit</span><span class="params">(<span class="keyword">const</span> <span class="keyword">int</span> &amp;x)</span></span>&#123;<span class="keyword">return</span> x&amp;-x;&#125;</span><br><span class="line"><span class="function"><span class="keyword">void</span> <span class="title">add</span><span class="params">(<span class="keyword">int</span> p)</span></span>&#123;<span class="keyword">for</span>(p++;p&lt;=n&lt;&lt;<span class="number">1</span>;p+=lowbit(p)) bit[p]++;&#125;</span><br><span class="line"><span class="function"><span class="keyword">int</span> <span class="title">sum</span><span class="params">(<span class="keyword">int</span> p)</span></span>&#123;<span class="keyword">int</span> res=<span class="number">0</span>;<span class="keyword">for</span>(p++;p;p-=lowbit(p)) res+=bit[p];<span class="keyword">return</span> res;&#125;</span><br><span class="line"></span><br><span class="line"><span class="built_in">vector</span>&lt;<span class="keyword">int</span>&gt; check_validity(<span class="keyword">int</span> N,<span class="built_in">vector</span>&lt;<span class="keyword">int</span>&gt; X,<span class="built_in">vector</span>&lt;<span class="keyword">int</span>&gt; Y,<span class="built_in">vector</span>&lt;<span class="keyword">int</span>&gt; S,<span class="built_in">vector</span>&lt;<span class="keyword">int</span>&gt; T,<span class="built_in">vector</span>&lt;<span class="keyword">int</span>&gt; L,<span class="built_in">vector</span>&lt;<span class="keyword">int</span>&gt; R)</span><br><span class="line">&#123;</span><br><span class="line">    n=N;m=X.size();q=S.size();</span><br><span class="line">    <span class="keyword">for</span>(<span class="keyword">int</span> i=<span class="number">1</span>;i&lt;=m;i++)</span><br><span class="line">    &#123;</span><br><span class="line">        e[i].u=X[i<span class="number">-1</span>];e[i].v=Y[i<span class="number">-1</span>];</span><br><span class="line">        <span class="keyword">if</span>(e[i].u&gt;e[i].v) swap(e[i].u,e[i].v);</span><br><span class="line">        e[i].u++;e[i].v++;</span><br><span class="line">    &#125;</span><br><span class="line">    sort(e+<span class="number">1</span>,e+<span class="number">1</span>+m,[](E a,E b)&#123;<span class="keyword">return</span> a.u&gt;b.u;&#125;);</span><br><span class="line">    t1.build(min);</span><br><span class="line">    sort(e+<span class="number">1</span>,e+<span class="number">1</span>+m,[](E a,E b)&#123;<span class="keyword">return</span> a.v&lt;b.v;&#125;);</span><br><span class="line">    t2.build(max);</span><br><span class="line">    <span class="keyword">for</span>(<span class="keyword">int</span> i=<span class="number">1</span>;i&lt;=n;i++)</span><br><span class="line">        g[t1.in[i]].push_back(&#123;<span class="number">1</span>,t2.in[i],<span class="number">0</span>,<span class="number">0</span>&#125;);</span><br><span class="line">    <span class="keyword">for</span>(<span class="keyword">int</span> i=<span class="number">0</span>,s,t,l,r,x1,x2,y1,y2;i&lt;q;i++)</span><br><span class="line">    &#123;</span><br><span class="line">        s=S[i]+<span class="number">1</span>;t=T[i]+<span class="number">1</span>;</span><br><span class="line">        l=L[i]+<span class="number">1</span>;r=R[i]+<span class="number">1</span>;</span><br><span class="line">        tie(x1,x2)=t1.jump(s,l,geq);</span><br><span class="line">        tie(y1,y2)=t2.jump(t,r,leq);</span><br><span class="line">        g[x1<span class="number">-1</span>].push_back(&#123;<span class="number">2</span>,y1<span class="number">-1</span>,i,<span class="number">1</span>&#125;);</span><br><span class="line">        g[x1<span class="number">-1</span>].push_back(&#123;<span class="number">2</span>,y2,i,<span class="number">-1</span>&#125;);</span><br><span class="line">        g[x2].push_back(&#123;<span class="number">2</span>,y1<span class="number">-1</span>,i,<span class="number">-1</span>&#125;);</span><br><span class="line">        g[x2].push_back(&#123;<span class="number">2</span>,y2,i,<span class="number">1</span>&#125;);</span><br><span class="line">    &#125;</span><br><span class="line">    ans.resize(q);</span><br><span class="line">    <span class="keyword">for</span>(<span class="keyword">int</span> i=<span class="number">0</span>;i&lt;=n&lt;&lt;<span class="number">1</span>;i++)</span><br><span class="line">        <span class="keyword">for</span>(Event ev : g[i])</span><br><span class="line">        &#123;</span><br><span class="line">            <span class="keyword">if</span>(ev.ty==<span class="number">1</span>) add(ev.a);</span><br><span class="line">            <span class="keyword">else</span> ans[ev.b]+=ev.c*sum(ev.a);</span><br><span class="line">        &#125;</span><br><span class="line">    <span class="keyword">for</span>(<span class="keyword">int</span> i=<span class="number">0</span>;i&lt;q;i++)</span><br><span class="line">        <span class="keyword">if</span>(ans[i]&gt;<span class="number">0</span>) ans[i]=<span class="number">1</span>;</span><br><span class="line">    <span class="keyword">return</span> ans;</span><br><span 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